[en] I derive a simple identity involving a sum of Clebsch-Gordan coefficients with zero magnetic quantum numbers. (orig.)

[en] The generalized theory of angular momentum, including both discrete and continuous variables, is considered. New objects-mixed Clebsch-Gordan coefficients-which relate the bases of ordinary |jm> states and coherent states, are introduced. Explicit expressions for these coefficients are given, and a consistent method for constructing relations of the generalized theory is presented. The Wigner-Eckart theorem for the basis of coherent states is formulated. 12 refs

[en] The Lorentz group is used as the group of homographic transformations on the Riemann sphere. Its Lie algebra is shown to have a very simple interpretation with the aid of cross products and constellation formalism. This property is used to give a constellation description of the Clebsch-Gordan series for the product of two states of spin 1

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[en] The Clebsch-Gordan coefficients of SU(3) are useful in calculations involving baryons and mesons, as well as in calculations involving arbitrary numbers of quarks. For the latter case, one needs the coupling constants between states of nonintegral hypercharges. The existing published tables are insufficient for many such applications, and therefore this collection has been compiled. This report supplies the isoscalar factors required to reconstruct the Clebsch-Gordan coefficients for a large set of products of representations. 15 refs., 5 tabs

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[en] The known general formula for the Clebsch-Gordan coefficients of the 3-dimensional rotation group involves one summation which results in explicit summation-free expressions for the coefficients where either one of the angular momenta is the sum of the other two or the magnetic quantum number corresponding to one of the angular momenta takes its maximum value in magnitude. By using very different techniques, explicit expressions for the coefficients < j₁,0;j₂,0|j,0>,< j,1/2;j₂,-1/2|j,0> are also obtained, where the integral or half-integral nature of the j's is indicated by the magnetic quantum number involved. Here the expressions depend upon whether j₁+j₂+j is an even or an odd integer. For these coefficients, the magnetic quantum numbers involved take their minimum value in magnitude. By using the recursion relation for the coefficients of the form < j₁,m,j₂,-m,|j,0>, we can calculate these coefficients in terms of the above known ones provided we also have the explicit value of the coefficient < j₁,1;j₂,-1|j,0> where j₁+j₂+j is an odd integer. (The recursion relation for these coefficients in terms of < j₁,0;j₂,0|j,0> becomes a triviality since < j₁,0;j₂,0|j,0> vanishes when j₁⁺j₂⁺j is an odd integer.) Our main purpose in this paper is to give an explicit expression for the coefficients < j₁,1;j₂,-1|j,0> where j₁+j₂+j is an odd integer. We are able to arrive at our expression by using a complicated transformation between hypergeometric functions which seems to have been neglected so far. For the coefficients where the magnetic quantum numbers have their minimum value in magnitude, this transformed expression becomes summation free and we obtain explicit values of the three already known coefficients and the fourth so far unknown. We believe that further study of this transformation may be useful on its own because it provides a link between very different types of expressions. (author)

[en] The even-even nuclei with respect to their proton-neutron numbers such as 40 Ca, 56 Ni, 208 Pb form the major closed shells. The low energy excited levels of the nuclei around these core nuclei can be explained successfully on the base of Nuclear Shell Model. Energy levels comparable with experimental spectrum energies can be calculated simply by adding the valence nucleon-nucleon interaction defined by -force to the single-particle energy level obtained by means of nuclear mass combination. Using the empirical parameters in -interaction, the energies of the excited levels of other nuclei around the core nucleus can be predicted. For the nuclei around the 1f_7/2 closed shell, calculated energy levels are found to be in good agreement with experimentally observed levels. For the general use, 3-j (Vector coupling) coefficients were calculated in the range of 0 j 7

[en] A simple identity involving a sum of Clebsch-Gordan coefficients is derived with zero magnetic quantum numbers